Finding the equation of the intersection

[cdot9]

Homework Statement

Plane 1:2x-3y+8z=7
Plane 2:x-8y+z=14
Plane 3:5x-14y+17z=28

Homework Equations

N/A

The Attempt at a Solution

I took plane 1 and subtracted (plane 2)x2 to get
13y+6z=-21 (I will refer to this as equation 1)

Then I took (plane 2)x5 and subtracted plane 3 to get
-26y-12z=32
which simplifies to
13y+6z=-16 (I will refer to this as equation 2)

Then I was stumped because the two equations contradict each other and state 0=-5
which leads me to believe that there is no values of x,y,z that can satisfy all planes

although the answers in the book say
"consistent [14/13,-21/13,0] + t[-61/13,-6/13,1]"

I understand how they get the answer I just don't understand why, if equation 1 and 2 do not provide a valid statement why do they continue and find the parametric equations

like let z = t
13y=-6t -21
y=-6/13t -21/13
then substitute y into plane 2 to get
x-8(-6/13z - 21/13) + z=14 and simplifies
x=61/13z +14/13
then z=0+t
which gets the correct parametric equations which I tested by using different values of t and the points do satisfy all planes
I guess the real question I have is why do they chose "equation 1" instead of "equation 2" which would yield a different result if you were to use that to fidn the parametric equations, which I'm assuming the parametric equations don't work and why do you continue after the elimination of equation 1 and 2 which results in 0=-5.

 

[SammyS]

Homework Statement

Plane 1:2x-3y+8z=7
Plane 2:x-8y+z=14
Plane 3:5x-14y+17z=28

Homework Equations

N/A

The Attempt at a Solution

I took plane 1 and subtracted (plane 2)x2 to get
13y+6z=-21 (I will refer to this as equation 1)

Then I took (plane 2)x5 and subtracted plane 3 to get
-26y-12z=32

Hello cdot9. Welcome to PF !

5×14 - 28 = 42, not 32.

which simplifies to
13y+6z=-16 (I will refer to this as equation 2)

Then I was stumped because the two equations contradict each other and state 0=-5
which leads me to believe that there is no values of x,y,z that can satisfy all planes

although the answers in the book say
"consistent [14/13,-21/13,0] + t[-61/13,-6/13,1]"

I understand how they get the answer I just don't understand why, if equation 1 and 2 do not provide a valid statement why do they continue and find the parametric equations

like let z = t
13y=-6t -21
y=-6/13t -21/13
then substitute y into plane 2 to get
x-8(-6/13z - 21/13) + z=14 and simplifies
x=61/13z +14/13
then z=0+t
which gets the correct parametric equations which I tested by using different values of t and the points do satisfy all planes
I guess the real question I have is why do they chose "equation 1" instead of "equation 2" which would yield a different result if you were to use that to fidn the parametric equations, which I'm assuming the parametric equations don't work and why do you continue after the elimination of equation 1 and 2 which results in 0=-5.